Local Linearization - Computation of Reachable Sets

2.4 Computation of Reachable Sets

2.4.2 Local Linearization

x=f(x(t),y(t),u(t)), 0=g(x(t),y(t),u(t)),

Set of explicit ODEs Set of algebraic equations

(2.38)

with f : Rⁿ^x⁺ⁿ^y⁺ⁿ^u 7→ Rⁿ^x and g : Rⁿ^x⁺ⁿ^y⁺ⁿ^u 7→ Rⁿ^y. Here, the vector x ∈ Rⁿ^x includes the differential states variables, the vectory∈Rⁿ^y specifies the algebraic variables, and the vectoru∈Rⁿ^u considers the controllable inputs and/or input uncertainties. Note that any fully-implicit DAE can always be transformed into the semi-explicit form, furthermore, the index-1 property holds if and only if the Jacobian matrix of the algebraic equations is invertible (non-singular), that is

det

∂g(x(t),y(t),u(t))

∂y

6= 0, t >0. (2.39)

2.4.2 Local Linearization

As stated earlier, our approach is based on abstracting the set of semi-explicit DAEs (2.38) to LDIs for consecutive time intervals; that is to perform a local linearization within the interval τk := [tk, tk+1], wherek∈Nis the time step. We only use constant-size time intervalstk =k·tr, withtr∈R⁺ being the time increment. An extension, however, covering variable-size time steps can found in [54].

For a concise notation, we introduce the vectorz:= (x^Ty^Tu^T)^T ∈Rⁿ^z, and the linearization point zk := (x^T_k y_k^Tu^T_k)^T. Notice that the subscriptk denotes the time step; that is the linearization point is updated at each time interval.

Remark 2.5. Since the linearization of (2.38) causes additional errors, these errors are determined in an over-approximative manner and considered as additional uncertain inputs. By recomputing the linearization for each τ_k, the over-approximation of the exact reachable set remains small and accurate results are guaranteed.

The local linearization of the DAE system (2.38), at each time interval τk is performed by an infinite Taylor series expressed as follows:

Infinite number of terms offi(z)

, (2.40)

Infinite number of terms ofgi(z)

, (2.41)

with ˜z(t) :=z(t)−z_k and the subscriptidenoting thei-th coordinate. Here the infinite number of terms of the Taylor expansion can be over-approximated using the following proposition, see [22]:

Proposition 2.1. Lagrangian remainder: For the time intervalτk, suppose the vectorz(t) varies within an arbitrary setR^z(τ_k), then the Taylor series of the DAE system (2.38) at the linearization point z_k is expressed via a 1-st order Taylor expansion in addition to the remainder terms bounded within the so-called Lagrangian remainder

withL^xandL^y corresponding to the remainders of differential and algebraic variables, respectively, and the variableµtakes any value from the setζ(τk), which is expressed according to

ζ(τk) :=

(

αz+ (1−α)zk : 0≤αi≤1,z∈R^z(τk) )

2.4.2.1 Abstraction to Linear Differential Inclusions

Based on Prop. 2.1, we can rewrite the Taylor expansions (2.40) and (2.41) in state-space form

∀t∈τk : x(t) =˙ f(x(t),y(t),u(t)), (2.42)

Here the matrices Ak,Bk,Ck denote the system matrices of the linearized differential equations, and likewise,D_k,E_k,F_k specify the matrices of the linearized algebraic constraints with proper dimension.

Due to the index-1 property of the DAE system, it is guaranteed that the matrixF_k is non-singular, thus

y∈ −F_k⁻¹·(g(z_k) +D_kx˜+E_ku˜⊕L^y(τ_k)), (2.44) then inserting (2.44) into (2.42) yields

where L(τk) is the set of linearization errors. One can further simplify (2.45) by merging the set of uncertain inputs and the set of linearization errors together, that is

U˜(τk) :=w(zk)⊕B˜k(U⊕(−uk))⊕L(τk). (2.46) This allows the abstraction of the differential equations of the DAE system (2.38) by the following LDI

∀t∈τ_k: x(t)˙ ∈A˜_kx(t)˜ ⊕ U(τ˜ _k). (2.47) Remark 2.6. Note that the DAE system (2.38) is abstracted at each time interval τk. Additionally, the inclusion (2.47) encloses all possible trajectories of the nonlinear DAE system, as we consider the linearization errors via the set of Lagrangian remainder included within the set of uncertain inputs ˜U.

2.4.2.2 Reachable Set Computation of Linear Inclusions

With the abstraction of the DAE system by (2.47), we can now employ known techniques to computing reachable sets for linear systems. Analogously to the solution of linear-time invariant (LTI) systems de-scribed by state equations, the solution of (2.47) for the next time instanttk+1, based on the superposition principle, is well-known to be [56]

R^x(tk+1) =e^A^˜^k^t^rR^x(tk) andRp([0, t_r]) denote the homogenous and particular (inhomogeneous) solutions, respectively, ande^At^˜^r is the matrix exponential. Note that the subscriptxcorresponds to differential state variables.

In this thesis we describee^At^˜^r via a finite Taylor series, up to an order σ, in addition to the remainder that considers all higher-order terms [101]

e^A^˜^k^t^r = I+

where the set M([0, tr]) corresponds to the set over-approximating the remainder terms of the Taylor expansion of the matrix exponential.

Note that (2.48) is just the solution of the LDI at the time tk+1. However, we are mainly interested in the solution within the overall time intervalτk; thus we must enclose the reachable sets at the time instants t_k and t_k+1 in the least conservative way. The over-approximation of the reachable set within τk is obtained as suggested in [38]

τk∈[tk+1, tk] : R^x_h(τk)⊆conv with the operatorconv(·) returning the convex hull enclosure of two sets as in (2.29). Here the additional term C([0, tr]) is an uncertainty factor handling correction of the enclosure set accounting for bloating of the reachable set. The computation ofC([0, tr]) as in (2.52) is based on [6, Prop. 3.1] which employs interval arithmetics.

With the computation of the homogenous solution, it remains to express the effect of the uncertain input;

that is to consider the inhomogeneous solution of the LDI (2.48). First we consider the simple case, in

which we assume that the system matrix ˜Ak is invertible and the trajectories of the uncertain inputs ˜U remains constant over the intervalτk; thus the set enclosing the particular solution may be expressed by

R_p([0, t_r]) = ˜U(τ_k) Z tr

e^A^˜^k^(t^r^−t)dt= ˜U(τ_k)·A˜⁻¹_k

e^A^˜^k^(t^r^−t)

t_r

= ˜U(τk)·A˜⁻¹_k (e^A^˜^k^t^r−I).

However, the aforementioned assumptions do not hold in many practical situations. In the event that either the matrix ˜Ak is singular or the set ˜U(τk) varies over time, one has to over-approximate the particular solution according to the following proposition, see [6, 85]:

Proposition 2.2. Over-approximation of the LDI particular solution: Given the LDI described as in (2.47), then the over-approximation of its particular solution is given by

Rp([0, tr])⊂

i=0

|A˜ⁱ_k|t_rⁱ⁺¹ (i+ 1)! ·U˜

⊕

M·tr·U˜

. (2.53)

withMandσas the variables appearing in (2.49) to over-approximate the matrix exponential.

Remark 2.7. Note that in Prop. 2.2, it is assumed that the set ˜U(τk) contains the origin, which is not always the case and certain correction measures have to be applied to consider this case. Simply put, one needs to split the effect of ˜U(τk) into a constant uc corresponding to its center and another set ˜U∆

specifying the deviation from the centeru_c. 2.4.2.3 Algorithmic Realization

Combining all previous results, the reachable set of the LDI for consecutive time intervals can be com-puted, as illustrated in Fig. 2.12, according to the following steps:

(a) Compute the homogeneous reachable sets of the LDI at different time instants based on previously computed sets.

(b) Obtain and bloat the convex hull enclosure of the homogenous reachable sets. The enlargement of the convex hull is necessary for two reasons: first to over-approximate the reachable set for each time interval, and second to account for the effect of the inhomogeneous solution of the LDI.

(a) Reachable sets at different time steps (b) Convex enclousre and enlargement

Figure 2.12: Stepwise computation of reachable sets of linear differential inclusions expressed as in (2.47).

The aforementioned steps can be summarized as in Alg. 1. The algorithm computes the reachable set of a family of LTI systems enclosed by the LDI described as in (2.47), for the next point in time using previously computed reachable sets. First, through lines 1-3, the algorithm computes the remainder of the matrix exponentialM([0, t_r]) based on the number of Taylor termsσwhich is a user-defined variable.

Using the remainder terms, one can over-approximate the matrix exponentiale^At^˜ ^r to obtain the homoge-nous solution of the LDI, in addition to the correction termC([0, tr]) accounting for over-approximation of the reachable set within the interval τk, as illustrated in lines 4 and 5. Finally, to fully describe the reachable set for the next time instant and current time interval, the algorithm over-approximates the particular solution of the LDI according to Prop. 2.2, as specified in lines 6-8.

Remark 2.8. In order to obtain the reachable for the complete time-horizont∈[0, tf], withtf as the terminal time specified by the user, one has to continuously call Alg. 1 to obtain the reachable sets at different time instants until t > t_f. Afterwards the reachable set is assembled via the union of the computed sets, that isR([0, t_f]) :=Stf /tr

k=1 R(τ_k).

Algorithm 1 ReachNextStep

Require: Uncertain inputs ˜U(τ_k), the time incrementt_r, number of Taylor termsσof the matrix expo-nential, time step k, and the system matrix ˜A_k

Ensure: R^x(t_k+1),R^x(τ_k)

i! ⊕M([0, tr]) .Over-approximation of matrix exponential

3: C([0, tr])^(2.52)= Pσ

Im Dokument Control and Stability of Power Systems using Reachability Analysis (Seite 40-45)